Reports of heavy shelling in city that was first to fall to ISIL in 2014 as beleaguered residents struggle to get out.

Iraq's Prime Minister Haider Al-Abadi has announced the start of a major offensive to retake Fallujah from the Islamic State of Iraq and the Levant (ISIL) group, which was quickly followed by reports of heavy shelling in the city.

Sources in the city said that six civilians were killed in shelling on Monday.

"We are here to announce that the Iraqi flag will soon be flying in the sky of Fallujah," Abadi had said a day earlier.

"Today, we will tear down the black flags of these despicable strangers who abducted this dear city. The time has come to liberate Fallujah and the victory will be ours. ISIL has nowhere to go but to flee the city."

The government, though, is expected to face a tough fight to push ISIL, also known as ISIS, out of the city, which is just 50km west of Baghdad and has been under its control for more than two years.

Fallujah was the first Iraqi city to fall to ISIL in January 2014, six months before it swept through large parts of Iraq and neighbouring Syria, shocking much of the world and heralding a major expansion of its territory.

Asked about the assault, the US State Department and the Pentagon did not immediately comment.

Richard Weitz, an analyst from the Hudson Institute, said the US military did not see the battle for Fallujah as the most immediate priority.

"The American military has been pushing them to reconquer Mosul as the decisive battle, not Fallujah," he said.

Civilians trapped without aid

Abadi and his government have faced severe political pressure in recent weeks as supporters of powerful cleric Moqtada al-Sadr, and people from other groups, staged protests over the government's failure to approve anti-corruption reforms and maintain security in Baghdad.

The capital has been hit by a string of suicide and car bombs claimed by ISIL, killing hundreds of people by some estimates, including more than 200 last week.

"The series of bombings that have happened in Baghdad over the last couple of weeks have made it important for Baghdad to take down Fallujah," Douglas Ollivant, an analyst with the New America Foundation who worked on Iraq under US presidents Obama and Bush, said.

"They think that the logistics base for these car bombs goes back to Fallujah."

Earlier on Sunday, the army said residents of Fallujah should leave "through secured routes".

About 20 families tried to escape but half were intercepted by ISIL or killed by bombs that had been planted along the road by the group, the Reuters news agency reported, citing residents.

Sources in Fallujah say that ISIL had told locals to move to the outskirts of the city but that it was not allowing them to get out.

Any familes who could not leave should raise white flags to mark their locations, the army said.

Up to 75,000 civilians have been trapped inside the city, caught between ISIL and forces encircling the city, which include the Iraqi army and a coalition of Shia Muslim armed groups known as Hashid Shaabi.

The United Nations food agency, the WFP, has not been able to deliver aid since last September.

The city on the Euphrates River had a prewar population of about 300,000. Known as the "City of Minarets and Mother of Mosques", it was badly damaged in two assaults by the US army against suspected al-Qaeda fighters in 2004.